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Black-Scholes equationVariational formulation
LocalizationDiscretization
Non-smooth initial data
Computational Methods for Quant. Finance II
Finite difference and finite element methods
Lecture 4
Computational Methods for Quant. Finance II
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Black-Scholes equationVariational formulation
LocalizationDiscretization
Non-smooth initial data
Outline
Black-Scholes equation
Variational formulation
Localization
Discretization
Non-smooth initial data
Computational Methods for Quant. Finance II
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Black-Scholes equationVariational formulation
LocalizationDiscretization
Non-smooth initial data
From expectation to PDE
Goal: compute thevalue of European optionwith payoffg which isthe conditional expectation
V(t, x) = E
eRTt r(Xs)dsg(XT) | Xt=x
, (1)
whereX is the (unique) solution of the stochastic differentialequation (the dynamics of the underlying of the option)
dXt=b(Xt) dt + (Xt) dWt. (2)
It turns out: V(t, x) solves a PDE. We need the notion of theso-calledinfinitesimal generatorof the process X.
Computational Methods for Quant. Finance II
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Black-Scholes equationVariational formulation
LocalizationDiscretization
Non-smooth initial data
PropositionLetAdenote the differential operator which is, for functionsf C2(R) with bounded derivatives, given by
(Af)(x) := 1
22(x)xxf(x) + b(x)xf(x). (3)
Then, the processMt :=f(Xt) t0 (Af)(Xs)ds is a martingale
with respect to the filtration ofW.
The operator A is theinfinitesimal generatorof the process X,which solves the SDE
dXt=(Xt) dWt+ b(Xt) dt.
Computational Methods for Quant. Finance II
Bl k S h l i
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Black-Scholes equationVariational formulation
LocalizationDiscretization
Non-smooth initial data
We need a discounted version of the above Proposition.
Proposition
Letf C1,2(R R) with bounded derivatives in x, letAbe as in(3) and assume thatr C0(R) is bounded. Then the process
Mt:=eRt0r(Xs)dsf(t, Xt)
t0
eRs0 r(X)d(tf+Afrf)(s, Xs)ds
is a martingale with respect to the filtration ofW.
We now are able to link thestochastic representationof the optionprice
V(t, x) = E
eRT
t r(Xs)dsg(XT) | Xt=x
with aparabolic partial differential equation.
Computational Methods for Quant. Finance II
Bl k S h l ti
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Black-Scholes equationVariational formulation
LocalizationDiscretization
Non-smooth initial data
Theorem
LetV C1,2(J R) C0(J R) with bounded derivatives in xbe a solution of
tV + AV rV = 0 in J R,V(T, x) = g(x) in R,
(4)
withAas in (3). Then, V(t, x) can also be represented as
V(t, x) = E
eRT
t r(Xs)dsg(XT) | Xt=x
. (5)
RemarkThe converse of this Theorem is also true. AnyV(t, x) as in (5),which isC1,2(J R) C0(J R) with bounded derivatives in x,solves the PDE (4). This is known asFeynman-Kac Theorem.
Computational Methods for Quant. Finance II
Black Scholes equation
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Black-Scholes equationVariational formulation
LocalizationDiscretization
Non-smooth initial data
We apply the Feynman-Kac Theorem to theBlack-Scholes model.
In the Black-Scholes market with no dividends, the risky assetsspot-price is modelled by a geometric Brownian motion X, i.e., theSDE for this model is as in (2), with coefficients
b(x) =rx, (x) =x,
where >0 and r 0 denote the (constant)volatilitythe(constant)interest rate, respectively.
Therefore, the SDE is given by (use S instead ofX)
dSt=rStdt + StdWt,
with infinitesimal generator
A := 1
22s2ss+ rss. (6)
Computational Methods for Quant. Finance II
Black Scholes equation
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Black-Scholes equationVariational formulation
LocalizationDiscretization
Non-smooth initial data
The Black-Scholes PDE
We obtain: the discounted price of a European contract withpayoffg(s), i.e.,
V(t, s) = E[er(Tt)g(ST)|St =s],
is equal to a regular solution V(t, s) of the Black-Scholes equation
tV + 12
2s2ssV + rssV rV = 0 in [0, T) R+
V(T, s) = g(s) in R+.
The infinitesimal generator (and hence the Black-Scholes PDE)degeneratesat s= 0. Furthermore, the PDE isbackwardin time.
Computational Methods for Quant. Finance II
Black-Scholes equation
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Black-Scholes equationVariational formulation
LocalizationDiscretization
Non-smooth initial data
To obtain a nondegenerate equation, we switch to thelog-priceprocess Xt = log(St) which solves the SDE
dXt = r
1
22
dt + dWt.
The infinitesimal generator for this process has constantcoefficients:
ABS :=1
2
2xx+ r 1
2
2x. We furthermore change totime-to-maturity t T t, to
obtain a forward parabolic problem.
Computational Methods for Quant. Finance II
Black-Scholes equation
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Black Scholes equationVariational formulation
LocalizationDiscretization
Non-smooth initial data
The Black-Scholes PDE in log-price
Thus, by setting V(t, s) =:v(T t, log s), the BS equation (inreal price) satisfied by V(t, s) becomes theBS equationfor v(t, x)in log-price
tv ABSv+ rv = 0 in (0, T] R
V(0, x) = g(ex) in R, (7)
with
ABS :=1
2 2xx+
r 1
2 2
x.
We next study the variational formulation of (7).
Computational Methods for Quant. Finance II
Black-Scholes equation
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c Sc o s qu t oVariational formulation
LocalizationDiscretization
Non-smooth initial data
Outline
Black-Scholes equation
Variational formulation
Localization
Discretization
Non-smooth initial data
Computational Methods for Quant. Finance II
Black-Scholes equation
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qVariational formulation
LocalizationDiscretization
Non-smooth initial data
Denote by (, ) the L2(R)-inner product, i.e., (, ) := R dx.Denote by aBS(, ) :H1(R) H1(R) R thebilinear formassociated to the operator ABS
aBS(, ) :=1
2
2(, ) + (2/2 r)(, ) + r(, ). (8)
The variational formulation of the Black-Scholes equation (7)reads:
Find u L2(J; H1(R)) H1(J; L2(R)) such that
(tu, v) + aBS(u, v) = 0, v H1(R), a.e. in J, (9)
u(0) =u0,
whereu0(x) :=g(ex).
Computational Methods for Quant. Finance II
Black-Scholes equation
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Variational formulationLocalization
DiscretizationNon-smooth initial data
Ifu0 L2(R), then problem (9) admits a unqiue solution, since
a
BS
(, ) iscontinuousand satisfies aGarding inequalityonH
1
(R
).Proposition
There exist constantsCi =Ci(, r)> 0, i= 1, 2, 3, such thatthere holds for all, H1(R)
|aBS(, )| C1H1H1 , aBS(, ) C22H1C32L2 .
Proof.
|aBS(, )| 1
22L2
L2+ |2/2 r|L2
+rL2L2 C1(, r)H1H1 .
aBS(, ) = 1
222L2+ r
2L2 =
1
222H1+ (r
2/2)2L2
1
222H1 |r
2/2|2L2 .
Computational Methods for Quant. Finance II
Black-Scholes equation
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Variational formulationLocalization
DiscretizationNon-smooth initial data
Outline
Black-Scholes equation
Variational formulation
Localization
Discretization
Non-smooth initial data
Computational Methods for Quant. Finance II
Black-Scholes equationV i i l f l i
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Variational formulationLocalization
DiscretizationNon-smooth initial data
u0(x) =g(ex) L2(R) implies an unrealistic growth
condition on the payoffg. For example, the payoff of both theput g(ex) = max{0, K ex} and the callg(ex) = max{0, ex K}are not in L2(R).
We can weaken this assumption by reformulating the problemon a bounded domain (which we have to do anyway for
discretizing the problem) The unbounded domain R of the log price x= log s is
truncated to a bounded domainG. In terms of financialmodeling, this corresponds to approximating the option priceby aknock-out barrier option.
LetG= (R, R), R >0, be an open subset and let
G = inf{t 0 | Xt Gc}
be thefirst hitting timeof the complement set Gc = R\G by
X.Computational Methods for Quant. Finance II
Black-Scholes equationV i ti l f l ti
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Variational formulationLocalization
DiscretizationNon-smooth initial data
The price of a knock-out barrier option in log-price with payoff
g(e
x
) is given byvR(t, x)= E
er(Tt)g(eXT)1{T0, q 1 such that the payoff functiong: R+ R satisfiesg(s) C(s + 1)
q for alls R+. Then, thereexistC(T, ), 1, 2>0, such that
|v(t, x) vR(t, x)| C(T, )e1R+2|x|.
Computational Methods for Quant. Finance II
Black-Scholes equationVariational formulation
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Variational formulationLocalization
DiscretizationNon-smooth initial data
We see from this Theorem that vR v exponentially for afixedx as R .
The artificial zero Dirichlet barrier type conditions at x= Rarenotdescribing correctly the asymptotic behavior of the
price v(t, x) for large |x|. Since the barrier option price vR is a good approximation to v
for |x| R, R should be selected substantially larger thanthe values ofx of interest.
The barrier option price vR can again be computed as the solutionof a PDE provided some smoothness assumptions.
Computational Methods for Quant. Finance II
Black-Scholes equationVariational formulation
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Variational formulationLocalization
DiscretizationNon-smooth initial data
Theorem
LetvR(t, x) C1,2(J R) C0(J R) be a solution of
tvR+ ABSvR rvR= 0, (11)
on [0, T) G where the terminal and boundary condition given by
vR(T, x) =g(ex), x G, vR(t, x) = 0, on (0, T) G
c.
Then,vR(t, x) can also be represented as in (10).
Now, we can restate the problem (9) on the bounded domain:
Find uR L2(J; H10 (G)) H
1(J; L2(G)) such that
(tuR, v) + aBS(uR, v) = 0, v H
10(G), a.e. inJ, (12)
uR(0) =u0|G .
Computational Methods for Quant. Finance II
Black-Scholes equationVariational formulation
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Variational formulationLocalization
DiscretizationNon-smooth initial data
Outline
Black-Scholes equation
Variational formulation
Localization
Discretization
Non-smooth initial data
Computational Methods for Quant. Finance II
Black-Scholes equationVariational formulation
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7/24/2019 Quant Black Scholes
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Variational formulationLocalization
DiscretizationNon-smooth initial data
Finite difference discretization
We discretize the PDE (11) directly using finite differences onbounded domain with homogeneous Dirichlet boundary conditions.UsingxxvR(tm, xi) h
2(umi+1 2umi u
mi1),
xvR(tm, xi) (2h)1(um
i+1
um
i1
)we obtain the matrix problem:
Find um+1 RN such that for m= 0, . . . , M 1,I+ kGBS
um+1 =
I (1 )kGBS
um,
u0 =u0,
where GBS =2/2R +
2/2 rC+ rI, is given explicitly with
R :=h2tridiag
1, 2, 1
, C := (2h)1tridiag
1, 0, 1
.
Computational Methods for Quant. Finance II
Black-Scholes equationVariational formulation
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LocalizationDiscretization
Non-smooth initial data
Finite element discretization
We discretize (12) using the -scheme and the finite element spaceVN =S
1T H
10(G). Setting again u0(x) :=g(e
x) and assuminguniform mesh width h and constant time step k, we obtain thematrix problem:
Find um+1 RN such that for m= 0, . . . , M 1
(M+ kABS)um+1 = (M k(1 )ABS)um,
u0N =u0 ,
whereM
ij = (bj , bi)L2
(G) andABS
ij =aBS
(bj , bi). Using themethods we have developed in chapter 3, we findABS =2/2S +
2/2 r
B+ rM, explicitly with
S := h1tridiag
1, 2, 1
, B := 21tridiag
1, 0, 1
,
M
:= 6
1
htridiag
1, 4, 1
.Computational Methods for Quant. Finance II
Black-Scholes equationVariational formulation
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LocalizationDiscretization
Non-smooth initial data
Outline
Black-Scholes equation
Variational formulation
Localization
Discretization
Non-smooth initial data
Computational Methods for Quant. Finance II
Black-Scholes equationVariational formulation
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LocalizationDiscretization
Non-smooth initial data
Graded mesh
Know: achievable convergence rate of linear continuous FEM+ -scheme is u uNL2(J;L2(G))= O(h
2 + kr) providedthe initial data u0(x) =g(e
x) (the payoff) satisfies
u0 H2(G). Here, r= 1 for [0, 1] \ {1/2} and r= 2 for= 1/2 (constant time step k is used).
However, for put and call contracts (u0 H3/2(G)) or for
digital options (u0 H1/2(G)), this regularity assumption
is not satisfied, and we expect areduction of the rate ofconvergencew.r. to time, i.e. we expect a lower r.
To recover the optimal convergence rate for u0 Hs(G),
0< s
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LocalizationDiscretization
Non-smooth initial data
SetT = 1. Let : [0, 1] [0, 1] be agrading functionwhich isstrictly increasing and satisfies
C0([0, 1]) C1((0, 1)), (0) = 0, (1) = 1.
We define for MN
the graded mesh by the time points,
tm=m
M
, m= 0, 1, . . . , M .
It can be shown that we obtain again the optimal convergence rate
if(t) = O(t) where depends on r and s, =(r, s).We give an example for a European call and digital (or binary)option.
Computational Methods for Quant. Finance II
Black-Scholes equationVariational formulation
L li i
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LocalizationDiscretization
Non-smooth initial data
Example
We set strike K= 100, volatility = 0.3, interest rate r= 0.01,maturity T = 1. For the discretization we use M= O(N),= 1/2, R= 3 and apply the L2-projection for u0.
We measure the discrete L2
(J; L2
(G))-error defined byMm=1 kih
m22 where
m22 :=N
i=1
|u(tm, xi) uN(tm, xi)|2.
both with constant time steps and with graded time steps. We usethe grading factor = 3 for the call option g(s) = max{0, s K}and = 25 for the digital option g(s) ={s>K}.
Computational Methods for Quant. Finance II
Black-Scholes equationVariational formulation
L li ti
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LocalizationDiscretization
Non-smooth initial data
101
102
103
104
104
103
102
101
100
s = 2.0
Call option
Digital option
N
L2-error
101
102
103
104
105
104
103
102
101
100
s = 2.0
Call option
Digital option
N
L2-err
or
Computational Methods for Quant. Finance II